CHAPTER 4:
Scatterplots and
Correlation
ESSENTIAL STATISTICS
Second Edition
David S. Moore, William I. Notz, and Michael A. Fligner
Lecture Presentation
Chapter 4 Concepts
Explanatory and Response Variables
Displaying Relationships: Scatterplots
Interpreting Scatterplots
Measuring Linear Association: Correlation
Facts About Correlation
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Explanatory and Response Variables
Most statistical studies examine data on more than one variable. In
many of these settings, the two variables play different roles.
Response Variable, Explanatory Variable
A response variable measures an outcome of a study.
An explanatory variable may help explain or influence changes in a
response variable.
Note: In many studies, the goal is to show that changes in one or
more explanatory variables actually cause changes in a response
variable. However, other explanatory-response relationships don’t
involve direct causation.
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Scatterplot
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4
The most useful graph for displaying the relationship between two
quantitative variables is a scatterplot.
A scatterplot shows the relationship between two quantitative
variables measured on the same individuals.
The values of one variable appear on the horizontal axis, and the
values of the other variable appear on the vertical axis.
Each individual in the data appears as a point on the graph.
How to Make a Scatterplot
1. Decide which variable should go on each axis. If a distinction
exists, plot the explanatory variable on the x-axis and the
response variable on the y-axis.
2. Label and scale your axes.
3. Plot individual data values.
Scatterplot
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Example: Make a scatterplot of the relationship between body
weight and pack weight for a group of hikers.
Body weight (lb)
120
187
109
103
131
165
158
116
Backpack weight (lb)
26
30
26
24
29
35
31
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Interpreting Scatterplots
To interpret a scatterplot, follow the basic strategy of data analysis
from Chapters 1 and 2. Look for patterns and important departures
from those patterns.
How to Examine a Scatterplot
As in any graph of data, look for the overall pattern and for
striking departures from that pattern.
• You can describe the overall pattern of a scatterplot by the
direction, form, and strength of the relationship.
• An important kind of departure is an outlier, an individual
value that falls outside the overall pattern of the relationship.
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Interpreting Scatterplots
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Two variables have a positive association when as one variable
decreases, the other variable also decreases and vice versa. In statistics, a
perfect positive correlation is represented by the value +1.00
Two variables have a negative association when one variable increases
as the other decreases, and vice versa. In statistics, a perfect negative
correlation is represented by the value -1.00
Interpreting Scatterplots
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There is one possible outlier: the
hiker with the body weight of 187
pounds seems to be carrying
relatively less weight than are the
other group members.
Strength
Direction
Form
 There is a moderately strong, positive, linear relationship
between body weight and pack weight.
 It appears that lighter hikers are carrying lighter
backpacks.
Measuring Linear Association

A scatterplot displays the strength, direction, and form of the
relationship between two quantitative variables.
The correlation r measures the strength of the linear relationship
between two quantitative variables.
æ xi - x öæ yi - y ö
1
÷÷
r=
ç
÷çç
å
n -1 è sx øè sy ø
•
•
•
•
•
r is always a number between -1 and 1.
r > 0 indicates a positive association.
r < 0 indicates a negative association.
Values of r near 0 indicate a very weak linear relationship.
The strength of the linear relationship increases as r moves
away from 0 toward -1 or 1.
• The extreme values r = -1 and r = 1 occur only in the case of
a perfect linear relationship.
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Facts About Correlation
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1. Correlation makes no distinction between explanatory and response
variables.
2. r has no units and does not change when we change the units of
measurement of x, y, or both.
3. Positive r indicates positive association between the variables, and
negative r indicates negative association.
4. The correlation r is always a number between -1 and 1.
Cautions:
• Correlation requires that both variables be quantitative.
•
Correlation does not describe curved relationships between variables,
no matter how strong the relationship is.
•
Correlation is not resistant. r is strongly affected by a few outlying
observations.
•
Correlation is not a complete summary of two-variable data.
Correlation
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Chapter 4 Objectives Review
Define explanatory and response variables
Construct and interpret scatterplots
Calculate and interpret correlation
Describe facts about correlation
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